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Nash equilibria of games with a continuum of players

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  • Guilherme Carmona

Abstract

We characterize Nash equilibria of games with a continuum of players (Mas-Colell (1984)) in terms of approximate equilibria of large finite games. For the concept of ("; ") equilibrium in which the fraction of players not " optimizing is less than " we show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an ("n; "n) equilibria, with "n converging to zero. The same holds for " equilibrium in which almost all players are " optimizing provided that either players payoff functions are equicontinuous or players action space is finite. Furthermore, we give conditions under which the above results hold for all approximating sequences of games. In our characterizations, a sequence of finite games approaches the continuum game in the sense that the number of players converges to infinity and the distribution of characteristics and actions in the finite games converges to that of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance.

Suggested Citation

  • Guilherme Carmona, 2004. "Nash equilibria of games with a continuum of players," Nova SBE Working Paper Series wp466, Universidade Nova de Lisboa, Nova School of Business and Economics.
    • Handle: RePEc:unl:unlfep:wp466
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    References listed on IDEAS

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    1. Drew Fudenberg & David Levine, 2008. "Limit Games and Limit Equilibria," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 2, pages 21-39, World Scientific Publishing Co. Pte. Ltd..
    2. Khan, M. Ali & Rath, Kali P. & Sun, Yeneng, 1997. "On the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Journal of Economic Theory, Elsevier, vol. 76(1), pages 13-46, September.
    3. Barlo, Mehmet & Carmona, Guilherme, 2015. "Strategic behavior in non-atomic games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 134-144.
    4. Dubey, Pradeep & Mas-Colell, Andreau & Shubik, Martin, 1980. "Efficiency properties of strategies market games: An axiomatic approach," Journal of Economic Theory, Elsevier, vol. 22(2), pages 339-362, April.
    5. Hildenbrand, W & Mertens, J F, 1972. "Upper Hemi-Continuity of the Equilibrium-Set Correspondence for Pure Exchange Economies," Econometrica, Econometric Society, vol. 40(1), pages 99-108, January.
    6. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, University Library of Munich, Germany.
    7. Guilherme Carmona, 2004. "On the Existence of Pure Strategy Nash Equilibria in Large Games," Game Theory and Information 0412008, University Library of Munich, Germany.
    8. Novshek, William & Sonnenschein, Hugo, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 171-187, June.
    9. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
    10. Green, Edward J, 1984. "Continuum and Finite-Player Noncooperative Models of Competition," Econometrica, Econometric Society, vol. 52(4), pages 975-993, July.
    11. Wooders, Myrna & Selten, Reinhard & Cartwright, Edward, 2001. "Some First Results for Noncooperative Pregames: Social Conformity and Equilibrium in Pure Strategies," Economic Research Papers 269360, University of Warwick - Department of Economics.
    12. Mas-Colell, Andreu, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part I: Mixed strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 153-170, June.
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    Cited by:

    1. Robin Nicole & Peter Sollich, 2018. "Dynamical selection of Nash equilibria using reinforcement learning: Emergence of heterogeneous mixed equilibria," PLOS ONE, Public Library of Science, vol. 13(7), pages 1-37, July.
    2. Guilherme Carmona, 2004. "On the Existence of Pure Strategy Nash Equilibria in Large Games," Game Theory and Information 0412008, University Library of Munich, Germany.
    3. Robin Nicole & Peter Sollich, 2017. "Dynamical selection of Nash equilibria using Experience Weighted Attraction Learning: emergence of heterogeneous mixed equilibria," Papers 1706.09763, arXiv.org.
    4. Bodoh-Creed, Aaron, 2013. "Efficiency and information aggregation in large uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2436-2466.
    5. Daniel Lacker & Kavita Ramanan, 2019. "Rare Nash Equilibria and the Price of Anarchy in Large Static Games," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 400-422, May.
    6. Aaron Bodoh-Creed & Brent Hickman, 2016. "College Assignment as a Large Contest," Working Papers 2016-27, Becker Friedman Institute for Research In Economics.
    7. Jian Yang, 2021. "Analysis of Markovian Competitive Situations Using Nonatomic Games," Dynamic Games and Applications, Springer, vol. 11(1), pages 184-216, March.
    8. Bodoh-Creed, Aaron L. & Hickman, Brent R., 2018. "College assignment as a large contest," Journal of Economic Theory, Elsevier, vol. 175(C), pages 88-126.
    9. Yang, Jian, 2011. "Asymptotic interpretations for equilibria of nonatomic games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 491-499.
    10. Carmona, Guilherme & Podczeck, Konrad, 2009. "On the existence of pure-strategy equilibria in large games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1300-1319, May.
    11. Peter Helgesson & Bernt Wennberg, 2015. "The N-Player War of Attrition in the Limit of Infinitely Many Players," Dynamic Games and Applications, Springer, vol. 5(1), pages 65-93, March.
    12. Jian Yang, 2017. "A link between sequential semi-anonymous nonatomic games and their large finite counterparts," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(2), pages 383-433, May.
    13. Jian Yang, 2015. "A Link between Sequential Semi-anonymous Nonatomic Games and their Large Finite Counterparts," Papers 1510.06809, arXiv.org, revised Jun 2016.
    14. Guilherme Carmona, 2009. "Intermediate Preferences and Behavioral Conformity in Large Games," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 11(1), pages 9-25, February.
    15. Jian Yang, 2015. "Analysis of Markovian Competitive Situations using Nonatomic Games," Papers 1510.06813, arXiv.org, revised Apr 2017.

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    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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